| L(s) = 1 | − 2.23·5-s + (−2.23 − 1.41i)7-s + 5.65i·11-s − 4.47·13-s + 3.16i·17-s + 3.16i·19-s − 4·23-s + 5.00·25-s − 2.82i·29-s + 6.32i·31-s + (5.00 + 3.16i)35-s − 9.89i·37-s − 4.47·41-s + 1.41i·43-s + 9.48i·47-s + ⋯ |
| L(s) = 1 | − 0.999·5-s + (−0.845 − 0.534i)7-s + 1.70i·11-s − 1.24·13-s + 0.766i·17-s + 0.725i·19-s − 0.834·23-s + 1.00·25-s − 0.525i·29-s + 1.13i·31-s + (0.845 + 0.534i)35-s − 1.62i·37-s − 0.698·41-s + 0.215i·43-s + 1.38i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2896433498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2896433498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
| good | 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 3.16iT - 17T^{2} \) |
| 19 | \( 1 - 3.16iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 9.89iT - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 1.41iT - 43T^{2} \) |
| 47 | \( 1 - 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 9.48iT - 61T^{2} \) |
| 67 | \( 1 - 7.07iT - 67T^{2} \) |
| 71 | \( 1 - 1.41iT - 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 12.6iT - 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77003541645011171722771078274, −7.41816570338724371271272961094, −6.84947001116294025536688769246, −5.99311272480285650491998092623, −4.91462802490384412718353213946, −4.25640024554332225054050450321, −3.71253151417336640592208087826, −2.68053844332892998366971013893, −1.67829232253079414466782810524, −0.11998043303085641195770618318,
0.67000876403028149922132624699, 2.45379452252262860207170576032, 3.09164769135147385461561817574, 3.76537166971372340244400138171, 4.78123586625999750233090330331, 5.47677645822631811091583762449, 6.31022088027317081745240555654, 7.01466814274357750611833581427, 7.66445396549220074319729770430, 8.556175268292677744687583630011
